Engineering quick reference

40 topics, one line each, in the order Hone teaches them.

Hone is a place to practise a career, one idea a day. This sheet is the whole Engineering track at a glance: every idea it covers, in the order they are taught, one line each. It is a map rather than a lesson. Read opens the full explanation of an idea; Practise gives you a question on it. Both are free, and reading needs no account at all.

From units to a working design · Units and numbers

SI and US units, and converting between themA unit is part of the number. Converting is multiplying by a fraction that equals one, written so the old unit cancels. Read: Two systems of units, and carrying a number between them · Practise SI and US units, and converting between them

significant figuresA calculated number cannot be more precise than the roughest measurement that went into it, and its written digits should say so. Read: Significant figures: how much of the number you actually know · Practise significant figures

checking an equation by its unitsReplace every symbol in an equation with its units and cancel. If what is left is not the unit the answer needs, the equation or a conversion is wrong. Read: Let the units check the equation for you · Practise checking an equation by its units

force, mass and weightWeight is a force: W = m × g, with g = 9.81 m/s² in SI and 32.2 ft/s² in US customary. Read: Mass is how much stuff; weight is the pull on it · Practise force, mass and weight

density and specific weightDensity ρ is mass per volume, in kg/m³. Specific weight γ = ρ × g is weight per volume, in N/m³. Read: Density and specific weight: how heavy a volume is · Practise density and specific weight

From units to a working design · Statics

vectors and their componentsAny force can be replaced by a horizontal part F cos θ and a vertical part F sin θ, and parts along the same axis simply add. Read: A force has a direction, so split it into two you can add · Practise vectors and their components

the free-body diagramCut the body free of everything touching it, and at every cut draw the force the removed thing was putting on the body. Nothing else goes on the page. Read: The free-body diagram: the body alone, and every force on it · Practise the free-body diagram

equilibrium of a particleWhen a small body is at rest, the forces on it sum to zero in every direction: ΣF_x = 0 and ΣF_y = 0. Read: Equilibrium of a particle: the forces add to nothing · Practise equilibrium of a particle

moments and the leverMoment is force times the perpendicular distance from the pivot to the line of the force: M = F × d, in N·m. Read: A moment: how hard a force is trying to turn something · Practise moments and the lever

equilibrium of a rigid bodyA body that is not moving has ΣF_x = 0, ΣF_y = 0 and ΣM = 0 about any point. Three equations, so three unknowns can be found. Read: Equilibrium of a rigid body: forces balance and turning balances · Practise equilibrium of a rigid body

reactions of a simple beamFor a beam on two supports, a moment sum about one support gives the other reaction, and the vertical force sum gives the first. Read: What each end of a simple beam carries · Practise reactions of a simple beam

a truss by the method of jointsAt a pin joint the member forces and the load add to zero, so a joint with two unknown members is solved by ΣF_x = 0 and ΣF_y = 0. Assume tension; a negative answer is compression. Read: A truss, one joint at a time · Practise a truss by the method of joints

From units to a working design · Strength of materials

stress: force over areaStress is force divided by area, σ = F / A. In N/mm² it is megapascals, and that is the unit steel is rated in. Read: Stress: the force, spread over the area that carries it · Practise stress: force over area

strain: how much it stretchesStrain is the change in length divided by the original length, ε = δ / L. It has no unit. Read: Strain: how much it stretched, for its length · Practise strain: how much it stretches

Hooke's law and Young's modulusBelow yield, σ = E × ε. E is Young's modulus, the stiffness of the material: about 200 GPa for steel, 70 GPa for aluminium. Read: Hooke's law: stress and strain are proportional, and E is the ratio · Practise Hooke's law and Young's modulus

axial deformation, PL over AEA rod of length L and area A under axial load P, in a material of modulus E, stretches δ = P L / (A E). Read: How much a rod stretches: PL over AE · Practise axial deformation, PL over AE

factor of safetyFactor of safety is the material's strength divided by the working stress, FS = σ_strength / σ_working. Turned round, the allowable stress is strength over the factor. Read: Factor of safety: how far the material is from its limit · Practise factor of safety

shear stress in a pinA pin loaded across its axis carries shear stress τ = V / A over its cross-section. In double shear two sections share the load, so the stress halves. Read: Shear in a pin: the force trying to slice it · Practise shear stress in a pin

thermal expansionA bar warmed by ΔT grows by δ = α L ΔT. If it cannot grow, that same strain becomes a stress σ = E α ΔT. Read: Thermal expansion: heat makes it longer, and holding it makes it stressed · Practise thermal expansion

From units to a working design · Circuits

Ohm's lawVoltage pushes, resistance resists, current is what results: V = I × R. Read: Ohm's law: push, resistance, and what flows · Practise Ohm's law

electrical powerPower is volts times amps, P = V × I, in watts. With Ohm's law it is also I² R and V² / R. Read: Electrical power: how fast the circuit turns energy into heat or work · Practise electrical power

series and parallel resistanceResistors in series add: R = R_1 + R_2. In parallel the reciprocals add: 1/R = 1/R_1 + 1/R_2, and the result is smaller than the smallest branch. Read: Series adds; parallel shares · Practise series and parallel resistance

Kirchhoff's voltage lawGoing once around a closed loop, the rises through sources and the drops across resistors sum to zero: ΣV = 0. Read: Kirchhoff's voltage law: round any loop, the volts add to zero · Practise Kirchhoff's voltage law

the voltage dividerTwo resistors in series across V_in give V_out = V_in × R_2 / (R_1 + R_2) across R_2, the one you read across. Read: The voltage divider: two resistors share the volts by their size · Practise the voltage divider

energy in a capacitorA capacitor C charged to V holds charge Q = C V and energy E = ½ C V². Read: A capacitor stores energy, and the energy goes as the square of the volts · Practise energy in a capacitor

RMS of a sine waveFor a sine wave, V_rms = V_peak / √2, about 0.707 of the peak. Power and heating use rms, so meters and ratings are in rms. Read: RMS: the DC voltage that would heat the same wire the same amount · Practise RMS of a sine wave

From units to a working design · Fluids and thermo

pressure at depthIn a fluid at rest, pressure rises with depth as p = ρ g h. Only depth, density and g matter; the shape of the container does not. Read: Pressure at depth: the weight of the column above you · Practise pressure at depth

Pascal's principle and the hydraulic jackPressure in a confined fluid is the same at every point, so F_1 / A_1 = F_2 / A_2. A small push on a small piston becomes a big push on a big one. Read: Pascal's principle: the same pressure everywhere, so area multiplies force · Practise Pascal's principle and the hydraulic jack

continuity: flow is area times speedFlow rate is area times speed, Q = A × v, and along a pipe with no leaks A_1 v_1 = A_2 v_2. A narrower pipe means faster fluid. Read: Continuity: what flows in must flow out · Practise continuity: flow is area times speed

Bernoulli between two pointsBetween two points along a flow with no losses, p + ½ ρ v² + ρ g z is the same at both. Faster means lower pressure; higher means lower pressure; and a tank drains at v = sqrt(2 g h). Read: Bernoulli: pressure, speed and height trade with each other · Practise Bernoulli between two points

the ideal gas lawP V = n R T, with P in pascals, V in cubic metres, n in moles, T in kelvin and R = 8.314 J/(mol·K). Read: The ideal gas law: pressure, volume, amount and temperature in one line · Practise the ideal gas law

sensible heat, Q = m c ΔTHeating a mass m of a material with specific heat c by ΔT takes Q = m c ΔT. For water, c is about 4186 J/(kg·K). Read: Sensible heat: how much energy to change a temperature · Practise sensible heat, Q = m c ΔT

efficiency of a heat engineNo engine running between a hot source at T_h and a cold sink at T_c can beat η = 1 − T_c / T_h, with both temperatures in kelvin. Read: The ceiling on a heat engine · Practise efficiency of a heat engine

work and powerWork is force times the distance moved along the force, W = F × d, in joules. Power is work over time, P = W / t, in watts. One horsepower is 746 W. Read: Work is force through a distance; power is how fast · Practise work and power

From units to a working design · Engineering economics and on the job

simple and compound interestSimple interest is I = P × i × n. Compound interest grows the whole sum each period: F = P × (1 + i)ⁿ. Read: Interest: simple grows in a line, compound grows on itself · Practise simple and compound interest

present and future valueA sum P today becomes F = P (1 + i)ⁿ in n periods; a sum F due in n periods is worth P = F / (1 + i)ⁿ today. Read: Present and future value: money moved through time · Practise present and future value

payback periodPayback period is the cost divided by the net saving per year, when the saving is the same every year. Read: Payback: how long until the saving has repaid the cost · Practise payback period

costing a bill of materialsEach line is quantity times unit price; the material cost is the sum; labour and overhead are added on top. Read: Costing a bill of materials · Practise costing a bill of materials

tolerance stacksStack parts in a line and the nominals add, and so do the tolerances. Worst case is the plain sum; the statistical estimate is the root of the sum of squares. Read: Tolerance stacks: when the parts all sit at their limits · Practise tolerance stacks

a unit-conversion chain from a specA conversion chain is one multiplication per unit, each factor written as a fraction equal to one, so the old units cancel on paper before any number is touched. Read: A datasheet in one system, a design in another · Practise a unit-conversion chain from a spec